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A finite difference method for quasi-linear and nonlinear differential functional parabolic equations with Neumann's condition

100%

Sapa S.

tom Vol. 49, [Z] 1

83-106

Classical solutions of nonlinear second-order partial dierential functional equations of parabolic type with Neumann's condition are approximated in the paper by solutions of associated explicit dierence functional equations. The functional dependence is of the Volterra type. Nonlinear estimates of the generalized Perron type for given functions are assumed. The convergence and stability results are proved with the use of the comparison technique. These theorems in particular cover quasi-linear equations, but such equations are also treated separately. The known results on similar dierence methods can be obtained as particular cases of our simple result.

Difference methods for infinite systems of hyperbolic functional differential equations on the Haar pyramid

100%

Jaruszewska-Walczak D.

2004

tom Vol. 24, no. 1

85-96

We consider the Cauchy problem for infinite system of differential functional equations ∂tzk(t, x) = fk(t, x, z, ∂xzk(t, x)), k mem N. In the paper we consider a general class of difference methods for this problem. We prove the convergence of methods under the assumptions that given functions satisfy the nonlinear estimates of the Perron type with respect to functional variables. The proof is based on functional difference inequalities. We constructed the Euler method as an example of difference method.

Implicit difference schemes for quasilinear parabolic functional equations

100%

Matusik M.

2012

tom Vol. 45, nr 4

869-886

We present a new class of numerical methods for quasilinear parabolic functional differential equations with initial boundary conditions of the Robin type. The numerical methods are difference schemes which are implicit with respect to time variable. We give a complete convergence analysis for the methods and we show that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given functions with respect to functional variables. Results obtained in the paper can be applied to differential equations with deviated variables and to differential integral problems.

Uniqueness of solutions of a generalized Cauchy problem for a system of first order partial functional differential equations

100%

Netka M.

tom Vol. 29, no. 1

69-79

The paper is concerned with weak solutions of a generalized Cauchy problem for a nonlinear system of first order differential functional equations. A theorem on the uniqueness of a solution is proved. Nonlinear estimates of the Perron type are assumed. A method of integral functional inequalities is used.

Implicit difference methods for infinite systems of hyperbolic functional differential equations

75%

Szafrańska A.

tom Vol. 50, [Z] 1

73-86

The paper deal with classical solutions of initial boundary value problems for infinite systems of nonlinear differential functional equations. Two types of difference schemes are constructed. First we show that solutions of our differential problem can be approximated by solutions of infinite difference functional schemes. In the second part of the paper we proof that solutions of finite difference systems approximate the solutions of aur differential problem. We give a complete convergence analysis for both types of difference methods. We adopt nonlinear estimates of the Perron type for given functions with respect to the functional variable. The proof of the stability is based on the comparison technique. Numerical examples are presented.